In mathematics, a representation on coordinate rings is a representation of a group on coordinate rings of affine varieties.
Let X be an affine algebraic variety over an algebraically closed field k of characteristic zero with the action of a reductive algebraic group G. G then acts on the coordinate ring k [ X ] of X as a left regular representation: ( g ⋅ f ) ( x ) = f ( g − 1 x ) . This is a representation of G on the coordinate ring of X.
The most basic case is when X is an affine space (that is, X is a finite-dimensional representation of G) and the coordinate ring is a polynomial ring. The most important case is when X is a symmetric variety; i.e., the quotient of G by a fixed-point subgroup of an involution.
Let k [ X ] ( λ ) be the sum of all G-submodules of k [ X ] that are isomorphic to the simple module V λ ; it is called the λ -isotypic component of k [ X ] . Then there is a direct sum decomposition:
k [ X ] = ⨁ λ k [ X ] ( λ ) where the sum runs over all simple G-modules V λ . The existence of the decomposition follows, for example, from the fact that the group algebra of G is semisimple since G is reductive.
X is called multiplicity-free (or spherical variety) if every irreducible representation of G appears at most one time in the coordinate ring; i.e., dim k [ X ] ( λ ) ≤ dim V λ . For example, G is multiplicity-free as G × G -module. More precisely, given a closed subgroup H of G, define
ϕ λ : V λ ∗ ⊗ ( V λ ) H → k [ G / H ] ( λ ) by setting ϕ λ ( α ⊗ v ) ( g H ) = ⟨ α , g ⋅ v ⟩ and then extending ϕ λ by linearity. The functions in the image of ϕ λ are usually called matrix coefficients. Then there is a direct sum decomposition of G × N -modules (N the normalizer of H)
k [ G / H ] = ⨁ λ ϕ λ ( V λ ∗ ⊗ ( V λ ) H ) ,
which is an algebraic version of the Peter–Weyl theorem (and in fact the analytic version is an immediate consequence.) Proof: let W be a simple G × N -submodules of k [ G / H ] ( λ ) . We can assume V λ = W . Let δ 1 be the linear functional of W such that δ 1 ( w ) = w ( 1 ) . Then w ( g H ) = ϕ λ ( δ 1 ⊗ w ) ( g H ) . That is, the image of ϕ λ contains k [ G / H ] ( λ ) and the opposite inclusion holds since ϕ λ is equivariant.
Let v λ ∈ V λ be a B-eigenvector and X the closure of the orbit G ⋅ v λ . It is an affine variety called the highest weight vector variety by Vinberg–Popov. It is multiplicity-free.